| Summary: | We study the out-of-equilibrium properties of a classical integrable
non-relativistic theory, with a time evolution initially prepared with a finite
energy density in the thermodynamic limit. The theory considered here is the
Non-Linear Schrodinger equation which describes the dynamics of the
one-dimensional interacting Bose gas in the regime of high occupation numbers.
The main emphasis is on the determination of the late-time Generalised Gibbs
Ensemble (GGE), which can be efficiently semi-numerically computed on arbitrary
initial states, completely solving the famous quench problem in the classical
regime. We take advantage of known results in the quantum model and the
semiclassical limit to achieve new exact results for the momenta of the density
operator on arbitrary GGEs, which we successfully compare with ab-initio
numerical simulations. Furthermore, we determine the whole probability
distribution of the density operator (full counting statistics), whose exact
expression is still out of reach in the quantum model.
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